Nanoscale continuum calculation of basal dislocation core structures in graphite

In this work, we calculate the core structures of basal dislocations in graphite in a nanoscale continuum framework. The model consists of a stack of buffered Kirchhoff plates where the plates represent the covalent interactions within individual graphene sheets and the buffer layers represent the secondary interactions between them. In the mid-plane of the buffer layers, cohesive surfaces are introduced to account for the nonlinear deformations due to basal dislocations. The cohesive surface separation is governed by using an empirical 4-8 Lennard–Jones potential. Meanwhile, their relative shear sliding is governed by using a newly proposed empirical periodic stacking-fault potential. With these potentials, the core structures of full dislocations and partials are calculated and examined. It is shown that the full dislocations automatically split into partials that repel each other. The core sizes of individual partials, measured between peak stresses, are about 5?nm wide for the edge component and slightly narrower for the screw component. Since these sizes are about 10 times the lattice constant, they lend credence to our continuum model of basal dislocation cores in graphite. It is also shown that when the dislocations are densely packed on the same glide plane, i.e. in a pile-up, with spacing one to two times the core size, the split partials retain their individual identity with well-defined and well-separated stress peaks. Meanwhile, the membrane normal stresses in the graphene sheets rise considerably at the pile-up tips which, in turn, may provoke further deformation and damage modes such as kinking and delamination.     

Multiscale characterization of dislocation processes in Al 5754

Multiscale characterization was performed on an Al–Mg alloy, Al 5754 O-temper, including in situ mechanical deformation in both the scanning electron microscope and the transmission electron microscope. Scanning electron microscopy characterization showed corresponding inhomogeneity in the dislocation and Mg distribution, with higher levels of Mg correlating with elevated levels of dislocation density. At the nanoscale, in situ transmission electron microscopy straining experiments showed that dislocation propagation through the Al matrix is characterized by frequent interactions with obstacles smaller than the imaging resolution that resulted in the formation of dislocation debris in the form of dislocation loops. Post-mortem chemical characterization and comparison to dislocation loop behaviour in an Al–Cr alloy suggests that these obstacles are small Mg clusters. Previous theoretical work and indirect experimental evidence have suggested that these Mg nanoclusters are important factors contributing to strain instabilities in Al–Mg alloys. This study provides direct experimental characterization of the interaction of glissile dislocations with these nanoclusters and the stress needed for dislocations to overcome them.     

Multiscale modelling of dislocation/grain boundary interactions. II. Screw dislocations impinging on tilt boundaries in Al

The interaction of dislocations with grain boundaries (GBs) determines a number of important aspects of the mechanical performance of materials, including strengthening and fatigue resistance. Here, the coupled atomistic/discrete-dislocation (CADD) multiscale method, which couples a discrete dislocation continuum region to a fully atomistic region, is used to study screw-dislocations interacting with Σ3, Σ11, and Σ9 symmetric tilt boundaries in Al. The low-energy Σ3 and Σ11 boundaries absorb lattice dislocations and generate extrinsic grain boundary dislocations (GBDs). As multiple screw dislocations impinge on the GB, the GBDs form a pile-up along the GB and provide a back stress that requires increasing applied load to push the lattice dislocations into the GB. Dislocation transmission is never observed, even with large GBD pile-ups near the dislocation/GB intersection. Results are compared with experiments and previous, related simulations. The Σ9 grain boundary, composed from a more complex set of structural units, absorbs screw dislocations that remain localized, with no GBD formation. With increasing applied stress, new screw dislocations are then nucleated into the opposite grain from structural units in the GB that are nearby but not at the location where the original dislocation intersected the boundary. The detailed behaviour depends on the precise location of the incident dislocations and the extent of the pile-up. Transmission can occur on both Schmid and non-Schmid planes and can depend on the shear stresses on the GB plane. A continuum yield locus for transmission is formulated. In general, the overall dissociation and/or transmission behaviour is also determined by the Burgers vectors and associated steps of the primitive vectors of the grain boundary, and the criteria for dislocation transmission formulated by Lee et al . [Scripta Metall. 23 799 (1989); Phil. Mag. A 62 131 (1990); Metall. Trans. A 21 2437 (1990)] are extended to account for these factors.     

A computational approach towards modelling dislocation transmission across phase boundaries

To study the nanoscopic interaction between edge dislocations and a phase boundary within a two-phase microstructure the effect of the phase contrast on the internal stress field due to the dislocations needs to be taken into account. For this purpose a 2D semi-discrete model is proposed in this paper. It consists of two distinct phases, each with its specific material properties, separated by a fully coherent and non-damaging phase boundary. Each phase is modelled as a continuum enriched with a Peierls–Nabarro (PN) dislocation region, confining dislocation motion to a discrete plane, the glide plane. In this paper, a single glide plane perpendicular to and continuous across the phase boundary is considered. Along the glide plane bulk induced shear tractions are balanced by glide plane shear tractions based on the classical PN model. The model's ability to capture dislocation obstruction at phase boundaries, dislocation pile-ups and dislocation transmission is studied. Results show that the phase contrast in material properties (e.g. elastic stiffness, glide plane properties) alone creates a barrier to the motion of dislocations from a soft to a hard phase. The proposed model accounts for the interplay between dislocations, external boundaries and phase boundary and thus represents a suitable tool for studying edge dislocation–phase boundary interaction in two-phase microstructures.     

Asymptotic expressions for the nearest and furthest dislocations in a pile-up against a grain boundary

In 1965, Armstrong and Head explored the problem of a pile-up of screw dislocations against a grain boundary. They used numerical methods to determine the positions of the dislocations in the pile-up and they were able to fit approximate formulae for the locations of the first and last dislocations. These formulae were used to gain insights into the Hall–Petch relationship. More recently, Voskoboinikov et al. used asymptotic techniques to study the equivalent problem of a pile-up of a large number of screw dislocations against a bimetallic interface. In this paper, we extend the work of Voskoboinikov et al. to construct systematic asymptotic expressions for the formulae proposed by Armstrong and Head. The further extension of these techniques to more general pile-ups is also outlined. As a result of this work, we show that a pile-up against a grain boundary can become equivalent to a pile-up against a locked dislocation in the case where the mismatch across the boundary is small.     

Understanding acoustoplasticity through dislocation dynamics simulations

The acoustoplastic effect in metals is routinely utilised in industrial processes involving forming, machining and joining, but the underlying mechanism is still not well understood. There have been earlier suggestions that dislocation mobility is enhanced intrinsically by the applied ultrasound excitation, but in subsequent deliberations it is routinely assumed that the ultrasound merely adds extra stresses to the material without altering its dislocation density or intrinsic resistance to deformation. In this study, a dislocation dynamics simulation was carried out to investigate the interactions of dislocations under the combined influence of quasi-static and oscillatory stresses. Under such combined stress states, dislocation annihilation is found to be enhanced leading to larger strains at the same load history. The simulated strain evolution under different stress schemes also closely resembles certain previously obtained experimental observations. The discovery here goes far beyond the simple picture that the ultrasound effect is merely an added-stress one, since here, the intrinsic strain-hardening potency of the material is found to be reduced by the ultrasound, through its effect on enhancing dislocation annihilation.     

Strongly non-local modelling of dislocation transport and pile-up

The purpose of this work is the continuum modelling of transport and pile-up of infinite discrete dislocation walls driven by non-local interaction and external loading. To this end, the underlying model for dislocation wall interaction is based on the non-singular Peierls–Nabarro (PN) model for the dislocation stress field. For simplicity, attention is restricted to walls consisting of single-sign dislocations and to continuous wall distributions on a single glide plane. In this context, the influence of strongly non-local (SNL; long-range) interaction, and its approximation as weakly non-local (WNL; short-range) are studied in the context of interaction- and external-load-driven wall pile-up at a boundary. The pile-up boundary is modelled via a spatially dependent dislocation mobility which decreases to zero at the boundary. The pile-up behaviour predicted by the current SNL-based continuous wall distribution modelling is consistent with that predicted by discrete wall distribution modelling. Both deviate substantially from the pile-up behaviour predicted by WNL-based continuous wall distribution modelling. As such, it is clearly essential to account in continuum models for the intrinsic SNL character of the interaction between same-sign dislocations ‘close’ to the boundary. Gradient-based WNL ‘approximation’ of this interaction is not justified.     

Free energy change of a dislocation due to a Cottrell atmosphere

The free energy reduction of a dislocation due to a Cottrell atmosphere of solutes is computed using a continuum model. We show that the free energy change is composed of near-core and far-field components. The far-field component can be computed analytically using the linearized theory of solid solutions. Near the core the linearized theory is inaccurate, and the near-core component must be computed numerically. The influence of interactions between solutes in neighbouring lattice sites is also examined using the continuum model. We show that this model is able to reproduce atomistic calculations of the nickel–hydrogen system, predicting hydride formation on dislocations. The formation of these hydrides leads to dramatic reductions in the free energy. Finally, the influence of the free energy change on a dislocation’s line tension is examined by computing the equilibrium shape of a dislocation shear loop and the activation stress for a Frank–Read source using discrete dislocation dynamics.     

Atomic simulations to evaluate effects of stacking fault energy on interactions between edge dislocation and spherical void in face-centred cubic metals

In this study, molecular dynamics simulations were performed to elucidate the effects of stacking fault energy (SFE) on the physical interactions between an edge dislocation and a spherical void in the crystal structure of face-centred cubic metals at various temperatures and for different void sizes. Four different types of interaction morphologies were observed, in which (1) two partial dislocations detached from the void separately, and the maximum stress corresponded to the detachment of the trailing partial; (2) two partial dislocations detached from the void separately, and the maximum stress corresponded to the detachment of the leading partial; (3) the partial dislocations detached from the void almost simultaneously without jog formation; and (4) the partial dislocations detached from the void almost simultaneously with jog formation. With an increase in void size or SFE, the interaction morphology changed in the above-mentioned order. It was observed that the magnitude of the critical resolved shear stress (CRSS) and its dependence on the SFE were determined by these interaction morphologies. The value of the CRSS in the case of interaction morphology (1) is almost equal to an analytical one based on the linear elasticity by employing the Burgers vector of a single partial dislocation. The maximum value of the CRSS is also obtained by the analytical model with the Burgers vector of the two partial dislocations.     

Climb via vacancy diffusion of edge dislocations in 2D dislocation microstructures

The prevention of strength degradation of components is one of the great challenges in solid mechanics. In particular, at high temperatures material may deform even at low stresses, a deformation mode known as deformation creep. One of the microstructural mechanisms that governs deformation creep is dislocation motion due to the absorption or emission of vacancies, which results in motion perpendicular to the glide plane, called dislocation climb. However, the importance of the dislocation network for the deformation creep remains far from being understood. In this study, a climb model that accounts for the dislocation network is developed, by solving the diffusion equation for vacancies in a region with a general dislocation distribution. The definition of the sink strength is extended, to account for the contributions of neighbouring dislocations to the climb rate. The model is then applied to dislocation dipoles and dislocation pile-ups, which are dense dislocation structures and it is found that the sink strength of dislocations in a pile-up is reduced since the vacancy field is distributed between the dislocations. Finally, the importance of the results for modelling deformation creep is discussed.     

Effect of source strength on dislocation pileups in the presence of stress gradients

The behaviour of a dislocation pileup with a finite-strength source is investigated in the presence of various stress gradients within a continuum model where a free-dislocation region exists around the source. Expressions for dislocation density and stress field within the pileup are derived for the situation where there are first and second spatial gradients in applied stress. For a pileup configuration under an applied stress, yielding occurs when the force acting on the leading dislocations at the pileup tips reaches the obstacle strength, and at the same time, it is required that the source be at the threshold stress for dislocation production. A numerical methodology is presented to solve the underlying equations that represent the yielding conditions. The yield stress calculated for a pileup configuration is found to depend on stress gradients, obstacle spacing and source/obstacle strengths. It increases with increasing the first stress gradient, yet dependent on the second stress gradient. Furthermore, while the dependency of yield stress on the obstacle spacing intensifies with increasing the first stress gradient, it diminishes with an increase of second stress gradient. Therefore, the second stress gradient, as a newly introduced parameter, can provide a new physical insight into the size-dependent plasticity phenomena at small length scales.     

Kinetics of Grain Boundary Recovery in Deformed Polycrystals

Annealing kinetics are studied for nonequilibrium ensembles of dislocations occurring in grain boundaries during plastic deformation. Two types of dislocation ensembles are considered: 1) walls of sessile extrinsic grain boundary dislocations (EGBDs), which cause a change of the GB misorientation angle, and 2) arrays of glissile EGBDs having a Burgers vector tangential to the grain boundary plane. For both types similar exponential relationships are obtained for the relaxation of the average EGBD density, with approximately the same characteristic time proportional to the cube of grain size.     

Screw dislocation in semi-infinite non-local hexagonal medium

Abstract

Nonlocal stresses of a screw dislocation near a free surface in a semi-infinite hexagonal medium are investigated by a surface dislocation model. The nonlocal image force on the screw dislocation due to the existing free surface is also obtained. All classical singularities for the stress and image force are eliminated. The maxima of the stress and image force are evaluated. A zero point of the stress is found, which predicts that different states of the shear stress exist simultaneously near the dislocation. The appearance of a zero value at the free surface and a maximum of the dislocation image force can be used to explain the existence of the dislocation free zone.     

Tunneling of screw dislocations in α-Fe

A model of tunneling of 1/2 [111] screw dislocations in b.c.c. metals from sessile to glissile core configurations described by dislocation splittings is proposed and numerical estimates for α-Fe are derived. The critical shear stress of the order of 50 kp/mm² should be temperature independent up to 8 K.     

Effect of isotropic stress on dislocation bias factor in bcc iron: an atomistic study

The effect of externally applied stress on the dislocation bias factor (BF) in bcc iron has been studied using a combination of atomistic static calculations and finite element integration. Three kinds of dislocations were considered, namely, a₀/2〈1 1 1〉{1 1 0} screw, a₀/2〈1 1 1〉{1 1 0} edge and a₀〈1 0 0〉{0 0 1} edge dislocations. The computations reveal that the isotropic crystal expansion leads to an increasing or constant dislocation bias, depending on the Burgers vector and type of dislocation. On the other hand, compressive stress reduces the dislocation bias for all the dislocations studied. Variation of the dislocation BF depending on dislocation type and Burgers vector is discussed by analysing the modification of the interaction energy landscape and the capture efficiency values for the vacancy and self-interstitial atom.     

3C-SiC薄膜小角晶界附近位错核心的原子组态研究

用LaB₆灯丝200 kV高分辨透射电镜拍摄了有小角晶界的3C-SiC/(001)Si 薄膜的[110]高分辨电子显微像. 用像解卷技术把本不直接反映晶体结构的实验像转化为结构像. 首先, 从完整区的结构像中分辨开间距仅为0.109 nm的Si和C原子柱; 随后按赝弱相位物体近似像衬理论, 分析像衬随晶体厚度的变化规律, 辨认出Si和C原子; 进而在原子水平上得出小角晶界附近两个复合位错的核心结构, 构建了结构模型并计算了模拟像. 实验像与模拟像的一致程度验证了结构模型的正确性. 于是, 在已知完整晶体结构的前提下, 仅从一帧实验高分辨像出发, 推演出原子的种类和位错核心的原子组态. 还讨论了3C-SiC 小角晶界的形成与晶界附近出现复合位错的关系.     

The interaction between a screw dislocation and an interfacial cruciform crack and collinear linear cracks

The interaction between a screw dislocation and an interfacial cruciform crack and collinear linear cracks under loads at infinity was investigated. General solutions of complex potentials to this problem were derived by using complex potential theory. As illustrative examples, the closed form solution for a screw dislocation interacting with an interfacial cruciform crack and a linear crack is obtained. The stress intensity factor and critical stress intensity factor for dislocation emission are also calculated. The results show that the shielding effect increases with the increase of the shear modulus and the distance between the two cracks, but it decreases with the increase of dislocation azimuth and the distance between the dislocation and the cruciform crack tip. The critical loads at infinity for dislocation emission increase with the increment of the emission angle, the distance the two cracks and the vertical length of the cruciform crack.     

Towards a further understanding of dislocation pileups in the presence of stress gradients

This study is an essential complement and extension to the stress-gradient concept recently proposed by Hirth. An analytic method is presented for studying the behaviour of double-ended dislocation pileup in the presence of various stress gradients by solving a singular integral equation based on the continuous approximation of dislocations. Four special cases of double-ended pileup in the presence of stress gradients are discussed in detail. The corresponding dislocation distribution, the length of pileup, the total number of dislocations within the pileup and the force on the leading dislocations at the pileup ends are derived, respectively. It is shown that both the number of dislocations and the force on the leading dislocation in a pileup are sensitive to the relative magnitude of stress near the dislocation source and both are less than that in constant stress case. Of particular importance, it is indicated that the small-scaled materials subjected to a stress involving a gradient would be stronger than that under a constant stress. Applied to wire torsion and foil bending, the stress gradient model predicts an increase in the initial yielding, which is in reasonable agreement with the recent experimental data. The proposed stress gradient concept may provide a new physical insight into the size-dependent plasticity phenomena at small length scale.     

Dislocation motion in icosahedral quasicrystals at elevated temperatures: Numerical simulation

We have simulated shear deformation of an icosahedral model quasicrystal at elevated temperatures with molecular dynamics. The generation of a dislocation loop was studied with a new visualization technique and a critical stress almost as large as the theoretical shear strength was measured. Built-in dislocations started to move at a temperature-dependent critical stress lower by one order of magnitude. While at zero temperature the dislocation propagated intermittently by large jumps, its motion became viscous as temperature increased. The dislocation cores bulged considerably owing to pinning at obstacles inherent in the structure. A calculation of the energy of a Peierls-Nabarro dislocation moving rigidly through the sample allowed us to determine the dominant obstacles. The results are considered in relation to two different models of quasicrystalline plasticity.     

Description of dislocation cores

In this paper we explore the possibility of describing the dislocation core of a dislocation by a distribution of infinitesimal dislocations smooth enough to remove all singularities. The results may be extended straightforwardly to the case of anisotropic elasticity and can be used to calculate accurately contrast effects. It is shown that precipitation at dislocation cores may induce a glissile to sessile transformation.Dedicated to Dr. Frantiek Kroupa in honour of his 70^th birthday.